Let $m>2$, $\alpha\geqslant 0$ and $a_{1},\ldots,a_{m}>0$, and define a smooth function $P:{\mathbin{\mathbb{R}}}\rightarrow{\mathbin{\mathbb{R}}}$ by $P(0)=\alpha+a_{1}+\cdots+a_{m}$ and

$$P(x)=\textstyle\frac{1}{x^{2}}\bigl(e^{\alpha x^{2}}\prod_{k=1}^{m}(1+a_{k}x^{2})-1\bigl),\quad x\neq 0.$$

(2.9)

Define real numbers $\phi_{1},\ldots,\phi_{m}$ by

$$\phi_{k}=a_{k}\int_{-\infty}^{\infty}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\,,$$

For $k=1,\ldots,m$ define a function $z_{k}:{\mathbin{\mathbb{R}}}\rightarrow{\mathbin{\mathbb{C}}}$ by

$$z_{k}(y)={\rm e}^{i\psi_{k}(y)}\sqrt{a_{k}^{-1}+y^{2}},\;\>\text{where}\;\>\psi_{k}(y)=a_{k}\int_{-\infty}^{y}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\,.$$

Now write ${\boldsymbol{\phi}}=(\phi_{1},\ldots,\phi_{m})$, and define a submanifold $L_{\boldsymbol{\phi}}^{\alpha}$ in ${\mathbin{\mathbb{C}}}^{m}$ by

$$L_{\boldsymbol{\phi}}^{\alpha}=\bigl\{(z_{1}(y)x_{1},\ldots,z_{m}(y)x_{m}):y\in{\mathbin{\mathbb{R}}},\;x_{k}\in{\mathbin{\mathbb{R}}},\;x_{1}^{2}+\cdots+x_{m}^{2}=1\bigr\}.$$

Then $L_{\boldsymbol{\phi}}^{\alpha}$ is a closed, embedded Lagrangian diffeomorphic to ${\mathbin{\cal S}}^{m-1}\times{\mathbin{\mathbb{R}}}$ and satisfying $H=\alpha F^{\perp}$. If $\alpha>0$ it is an LMCF expander, and if $\alpha=0$ it is one of the Lawlor necks $L_{{\boldsymbol{\phi}},A}$ from Example 2.5. It is graded, with Lagrangian angle

$$\theta_{L_{\boldsymbol{\phi}}^{\alpha}}\bigl((z_{1}(y)x_{1},\ldots,z_{m}(y)x_{m})\bigr)=\textstyle\sum_{k=1}^{m}\psi_{k}(y)+\arg\bigl(-y-iP(y)^{-1/2}\bigr).$$

Note that the only difference between the constructions of $L_{{\boldsymbol{\phi}},A}$ in Example 2.5 and $L_{\boldsymbol{\phi}}^{\alpha}$ above is the term $e^{\alpha x^{2}}$ in (2.9), which does not appear in (2.3). If $\alpha=0$ then $e^{\alpha x^{2}}=1$, and the two constructions agree.

As in [43, Th. D], $L_{\boldsymbol{\phi}}^{\alpha}$ is asymptotically conical, with cone $C$ the union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of two Lagrangian $m$-planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ given by

$$\Pi_{0}=\bigl\{(x_{1},\ldots,x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},\;\>\Pi_{\boldsymbol{\phi}}=\bigl\{({\rm e}^{i\phi_{1}}x_{1},\ldots,{\rm e}^{i\phi_{m}}x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\}.$$

But in contrast to Example 2.5, for $\alpha>0$ we do not have $\phi_{1}+\cdots+\phi_{m}=\pi$, so $\Pi_{\boldsymbol{\phi}}$ and $C$ are not special Lagrangian.

In [43, Th. D] we prove that for fixed $\alpha>0$, the map $\Phi^{\alpha}:(a_{1},\ldots,a_{m})\mapsto(\phi_{1},\ldots,\phi_{m})$ gives a diffeomorphism

$$\Phi^{\alpha}:(0,\infty)^{m}\longrightarrow\bigl\{(\phi_{1},\ldots,\phi_{m})\in(0,\pi)^{m}:0<\phi_{1}+\cdots+\phi_{m}<\pi\bigr\}.$$

That is, for all $\alpha>0$ and ${\boldsymbol{\phi}}=(\phi_{1},\ldots,\phi_{m})$ with $0<\phi_{1},\ldots,\phi_{m}<\pi$ and $0<\phi_{1}+\cdots+\phi_{m}<\pi$, the above construction gives a unique LMCF expander $L_{\boldsymbol{\phi}}^{\alpha}$ asymptotic to $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$.

In dimension $m=1$, the unique connected Lagrangian MCF translator in ${\mathbin{\mathbb{C}}}$, up to rigid motions and rescalings, is the ‘grim reaper’

$$\bigl\{x+iy\in{\mathbin{\mathbb{C}}}:y\in(-\pi/2,\pi/2),\quad x=-\log\cos y\bigr\},$$

with translating vector $v=1\in{\mathbin{\mathbb{C}}}$, which is sketched in Figure 2.1.

For given constants $\alpha>0$ and $a_{1},\ldots,a_{m-1}>0,$ define

$$\psi_{j}(y)=\int_{-\infty}^{y}\frac{{\rm d}t}{(\frac{1}{a_{j}}+t^{2})\sqrt{P(t)}}\,,\;\>\text{where}\;\>P(t)=\frac{1}{t^{2}}\bigg(\prod_{k=1}^{m-1}(1+a_{k}t^{2})e^{\alpha t^{2}}-1\bigg),$$

for $j=1,\ldots,m-1$ and $y\in{\mathbin{\mathbb{R}}}$. Then

	$\displaystyle L=$	$\displaystyle\bigl\{\bigl(x_{1}\textstyle\sqrt{\frac{1}{a_{1}}\!+\!y^{2}}\,e^{i\psi_{1}(y)},\ldots,x_{m-1}\sqrt{\frac{1}{a_{m-1}}\!+\!y^{2}}\,e^{i\psi_{m-1}(y)},\textstyle{\textstyle\frac{1}{2}}y^{2}\!-\!{\textstyle\frac{1}{2}}\sum_{j=1}^{m-1}x_{j}^{2}$
		$\displaystyle-\textstyle\frac{i}{\alpha}\sum_{j=1}^{m-1}\psi_{j}(y)-\textstyle\frac{i}{\alpha}\arg(y+iP(y)^{-1/2})\bigr):x_{1},\ldots,x_{m-1},y\in{\mathbin{\mathbb{R}}}\bigr\}$		(2.10)

is a closed, embedded Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ diffeomorphic to ${\mathbin{\mathbb{R}}}^{m},$ which is a Lagrangian MCF translator with translating vector $(0,\ldots,0,\alpha)\in{\mathbin{\mathbb{C}}}^{m}$.

Define $\phi_{1},\ldots,\phi_{m-1}\in{\mathbin{\mathbb{R}}}$ by

$$\phi_{j}=\int_{-\infty}^{\infty}\frac{{\rm d}t}{(\frac{1}{a_{j}}+t^{2})\sqrt{P(t)}}\,.$$

Then $\phi_{1},\ldots,\phi_{m-1}\in(0,\pi)$ with $\phi_{1}+\cdots+\phi_{m-1}<\pi$, and $\psi_{j}(y)\rightarrow\phi_{j}$ as $y\rightarrow\infty$, and $\psi_{j}(y)\rightarrow 0$ as $y\rightarrow-\infty$. For fixed $\alpha>0,$ the map $(a_{1},\ldots,a_{m-1})\mapsto(\phi_{1},\ldots,\phi_{m-1})$ is a 1-1 correspondence from $(0,\infty)^{m-1}$ to $\bigl\{(\phi_{1},\ldots,\phi_{m-1})\in(0,\pi)^{m-1}:\phi_{1}+\cdots+\phi_{m-1}<\pi\bigr\}$.

The phase function $\theta_{L}$ of $L$ in (2.10) is a monotone decreasing function of $y$ only, with limits $\pi$ as $y\rightarrow-\infty$ and $\sum_{j=1}^{m-1}\phi_{j}$ as $y\rightarrow+\infty$. Thus, by choosing $\sum_{j=1}^{m-1}\phi_{j}$ close to $\pi,$ the phase variation of $L$ can be made arbitrarily small.

We can give the following heuristic description of $L$ in (2.10). If $y\gg 0$ then $\psi_{j}(y)\approx\phi_{j}$ and $\sqrt{\frac{1}{a_{j}}+y^{2}}\approx y$, and the terms $-\frac{i}{\alpha}\sum_{j=1}^{n}\psi_{j}(y)-\frac{i}{\alpha}\arg(y+iP(y)^{-1/2})$ are negligible compared to ${\textstyle\frac{1}{2}}y^{2}$ in the last coordinate. Thus, the region of $L$ with $y\!\gg\!0$ is in a weak sense approximate to

$$\bigl\{\bigl(x_{1}ye^{i\phi_{1}},\ldots,x_{m-1}ye^{i\phi_{m-1}},\textstyle{\textstyle\frac{1}{2}}y^{2}-{\textstyle\frac{1}{2}}\sum_{j=1}^{m-1}x_{j}^{2}\bigr):x_{1},\ldots,x_{m-1}\in{\mathbin{\mathbb{R}}},\;y>0\bigr\}.$$

But this is just an unusual way of parametrizing

$$\Pi_{\boldsymbol{\phi}}=\bigl\{\bigl(y_{1}e^{i\phi_{1}},\ldots,y_{m-1}e^{i\phi_{m-1}},y_{m}\bigr):y_{j}\in{\mathbin{\mathbb{R}}}\bigr\}\setminus\bigl\{(0,\ldots,0,y_{m}):y_{m}\leqslant\penalty 0\bigr\},$$

the complement of a ray in a Lagrangian plane. Similarly, the region of $L$ with $y\ll 0$ is in a weak sense approximate to

$$\Pi_{0}=\bigl\{(y_{1},\ldots,y_{m-1},y_{m}):y_{j}\in{\mathbin{\mathbb{R}}}\bigr\}\setminus\bigl\{(0,\ldots,0,y_{m}):y_{m}\leqslant\penalty 0\bigr\}.$$

So, $L$ can be roughly described as asymptotic to the union of two Lagrangian planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}\cong{\mathbin{\mathbb{R}}}^{m}$ which intersect in an ${\mathbin{\mathbb{R}}}$ in ${\mathbin{\mathbb{C}}}^{m}$, the $y_{m}$-axis $\bigl\{(0,\ldots,0,y_{m}):y_{m}\in{\mathbin{\mathbb{R}}}\bigr\}$. To make $L$, we glue these Lagrangian planes by a kind of ‘connect sum’ along the negative $y_{m}$-axis $\bigl\{(0,\ldots,0,y_{m}):y_{m}\leqslant\penalty 0\bigr\}$. Under Lagrangian mean curvature flow, $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ remain fixed, but the gluing region translates in the positive $y_{m}$ direction, as though $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ are being ‘zipped together’.

A slightly more accurate description of the ends of $L$ for large $y$ is that $L$ approximates $\tilde{\Pi}_{\boldsymbol{\phi}}$ when $y\gg 0$ and $\tilde{\Pi}_{0}$ when $y\ll 0$, where $\tilde{\Pi}_{\boldsymbol{\phi}}$ and $\tilde{\Pi}_{0}$ are the non-intersecting affine Lagrangian planes in ${\mathbin{\mathbb{C}}}^{m}$

$$\begin{split}\tilde{\Pi}_{\boldsymbol{\phi}}&=\bigl\{\bigl(y_{1}e^{i\phi_{1}},\ldots,y_{m-1}e^{i\phi_{m-1}},y_{m}\!-\!\textstyle\frac{i}{\alpha}(\phi_{1}\!+\!\cdots\!+\!\phi_{m-1})\bigr):y_{j}\!\in\!{\mathbin{\mathbb{R}}}\bigr\},\\ \tilde{\Pi}_{0}&=\bigl\{\bigl(y_{1},\ldots,y_{m-1},y_{m}-\textstyle\frac{i\pi}{\alpha}\bigr):y_{j}\in{\mathbin{\mathbb{R}}}\bigr\}.\end{split}$$

(2.11)

We will discuss these Lagrangian MCF translators further in Example 3.32.